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Approximation of quasi-stationary distributions for 1-dimensional killed diffusions with unbounded drifts
Denis Villemonais
To cite this version:
Denis Villemonais. Approximation of quasi-stationary distributions for 1-dimensional killed diffusions with unbounded drifts. 2009. �hal-00387054�
Approximation of quasi-stationary distributions for 1-dimensional killed diffusions with unbounded drifts
Denis Villemonais May 22, 2009
Abstract
The long time behavior of an absorbed Markov process is well described by the limiting distribution of the process conditioned to not be killed when it is observed.
Our aim is to give an approximation’s method of this limit, when the process is a 1-dimensional Itˆo diffusion whose drift is allowed to explode at the boundary. In a first step, we show how to restrict the study to the case of a diffusion with values in a bounded interval and whose drift is bounded. In a second step, we show an approximation method of the limiting conditional distribution of such diffusions, based on a Fleming-Viot type interacting particle system. We end the paper with two numerical applications : to the logistic Feller diffusion and to the Wright-Fisher diffusion with values in ]0,1[ conditioned to be killed at 0.
Key words : quasi-stationary distribution, interacting particle system, empirical process, Yaglom limit, diffusion process.
MSC 2000 subject : Primary 65C50, 60K35; secondary 60J60
1 Introduction
Let (Xt) be a killed Markov process with law P, taking its values in E∪ {∂}, where ∂ is a cemetery point. We denote by τ∂ = inf{t ≥ 0, Xt = ∂} the killing time of (Xt).
A probability measure ν on E is called a quasi-stationary distribution (QSD) if, for all t ≥ 0, the distribution of the process X, initially distributed with respect to ν and conditioned to be not killed before time t, is stillν at timet, that is Pν(Xt∈A|τ∂ > t) = ν(A) for every A ⊂ E and t ≥ 0. Without loss of generality, we suppose that ∂ is an absorbing point, so that {τ∂ > t}={Xt 6=∂}.
Letµ be a probability measure on E. If it exists and provided it is a probability, the limiting conditional distribution
t→+∞lim
Px(Xt∈.|Xt6=∂)
is called the Yaglom limit for µ, from the Russian Mathematician A.M. Yaglom. He showed in [27] that the limiting conditional distribution of the number of descendants in the nth generation of a Galton-Watson process always exists in the subcritical case.
The existence or uniqueness of such invariant conditional distributions have been proved in a host of contexts. When E is finite, it is proved in [7] that there exists a unique QSD ν and that the Yaglom limit converges to ν independently of the initial distribution. In [4], the case of a birth and death process on N is studied. For this pro- cess, the set of QSDs is either empty, or a singleton, or a continuum indexed by a real parameter and given by an explicit recursive formula. This is an exception : most of the known results on QSDs are related with existence or uniqueness problems. In [11], the existence of a quasi-stationary distribution for a continuous time Markov chain on N killed at 0 is proved under conditions on moments of the killing time, using an original renewal dynamical approach. In [6], the case of 1-dimensional diffusion on [0,+∞[ with C1 drift and killed at 0 are studied, with the assumption that +∞is a natural boundary.
The dependence between the initial measure and the Yaglom limit is explored in [19] (for a Brownian motion with constant drift killed at 0) and [18] (for the Orstein-Uhlenbeck process killed at 0). In [26], the case of 1-dimensional diffusions with general killing on the interior of a given interval is investigated. In [3], the authors study the existence and uniqueness of the QSD for 1-dimensional diffusions killed at 0 and whose drift is allowed to explode at the boundary, which is the case under study in the present paper. See [23]
for a regularly updated extensive bibliography on QSD.
In this paper we are concerned with 1-dimensional Itˆo diffusions with values in ]0,+
∞[∪{∂} killed at 0 and defined by the stochastic differential equation dXt=dBt−q(Xt)dt, X0 =x >0,
where B is a standard 1-dimensional Brownian motion and q ∈C1(]0,+∞[). In [3], the Yaglom limit of this process is studied and the authors give some conditions on the drift q, which are sufficient for the existence and the uniqueness of the QSD. In particular, they allow the drift to explode at the origin. As explained in the paper, this diffusion is closely related with some Markov mortality models. Such applications need the computation of the process QSD, but the tools used in [3] are based on spectral theory’s arguments and don’t allow us to get explicit values. Our aim is to give an easily simulable approximation’s method of this QSD.
The problem of QSD’s approximation has been already explored in [2], [12] whenE is a bounded open set of Rd and X is a Brownian motion killed at the boundary of E. The authors proved an approximation’s method exposed in [1], which is based on a Fleming- Viot type system of interacting particles whose number is going to infinity. In [10], it is proved that this method works well for a continuous time Markov chain in a countable state space under suitable assumption on the transition’s rates (moreover, the existence of a QSD is a consequence of the approximation’s method). New difficulties arise from our case with unbounded drift. For instance, the interacting particle process introduced in [1] isn’t necessarily well defined. To avoid this difficulty, we begin by proving that one can approximate our QSD by the QSD’s of diffusions with bounded drifts.
Let us denote byPǫthe law of a diffusion with values in ]ǫ,1/ǫ[, defined by the stochastic differential equation dXt = dBt−q(Xt)dt and killed when it hits ǫ or 1/ǫ. In [22], it is proved that the Yaglom limit associated with Pǫ exists and is its unique QSD. We will denote it byνǫ. In the first part of this paper, we give some conditions onq ∈C1(]0,+∞[) for the family (νǫ)0<ǫ≤1/2 to be tight and to converge, whenǫ →0, to a QSD for the law P0. We point out the fact that this result remains valid in the case of an unbounded drift
diffusion with values in a bounded interval. In a second part, we prove an approximation method for each probability measureνǫ, based on the interacting process introduced in [1].
Fix ǫ >0 and let us describe the interacting particle process of size N ≥2: each particle moves independently in ]ǫ,1/ǫ[, each one with law Pǫ until one of them hits the boundary.
At this time, the killed particle jumps on the position of an other particle, chosen uniformly between the N −1 remaining one. Then the particles evolve independently, until one of them is killed and so on (see Figure 1). One has to prove that the particles don’t degenerate
1
τ4 τ3
τ2
0 τ1 t
X2 X1
Figure 1: The interacting particle system (X1,X2)
at the boundary. In [2], the authors prove a non-degeneracy result with arguments based on a construction of the d-dimensional Brownian motion due to Itˆo, where d ≥ 2. It seems that this tool can’t be easily generalized to other diffusions. To prove such results under our settings, we build an original coupling between the interacting particle process and an independent particle system of the same size. This coupling is valid for all drifted Brownian motions with continuous bounded drift, killed at the boundary of a bounded interval of R+. It will be used in each step of the proof.
We conclude the paper by two numerical applications. At first, we treat the case of the logistic Feller diffusion introduced in [17] and studied in [3] with values in ]0,+∞[, driven by the stochastic differential equation
dZt=p
ZtdBt+ (rZt−cZt2)dt, Z0 =z >0,
where B is a 1-dimensional Brownian motion and r, care two positive constants. Clearly, 0 is an absorbing state for this diffusion. In a second time we study in detail the case of the Wright-Fisher diffusion on ]0,1[ conditioned to be killed at 1 (see [13]). This diffusion takes values in ]0,1[, and is defined by
dZt=p
Zt(1−Zt)dBt+ (1−Zt)dt, Z0 =z ∈]0,1[,
where B is a 1-dimensional Brownian motion. This diffusion is absorbed at 1.
2 From unbounded drift to bounded drift
Let P0
x be the law of a diffusion process taking its values in ]0,+∞[∪{∂}, killed when it hits 0 and defined by the stochastic differential equation (SDE)
dXt=dBt−q(Xt)dt, X0 =x >0,
whereB is a 1-dimensional Brownian motion. The driftq is taken in the set of real valued continuously differentiable functions C1(]0,+ ∞[). We denote by L0 the infinitesimal generator associated with P0.
We define,∀x∈]0,+∞[,
Q(x) = Z x
1
q(y)dy, dµ(x) = e−2Q(x)dx, and
W(x) =q(x)2 −q′(x).
For all ǫ ∈]0,1/2[, we define Pǫx as the law of the diffusion taking its values in ]ǫ,1/ǫ[, defined by the SDE
dXt=dBt−q(Xt)dt, X0 =x∈]ǫ,1/ǫ[
and killed when it hits the boundary{ǫ,1/ǫ}. LetLǫbe the infinitesimal generator 1/2∆− q∇with the Dirichlet boundary condition on{ǫ,1/ǫ}. −Lǫ has a simple real eigenvalueλǫ
(see [22, Theorem KR]) at the bottom of its spectrum. The corresponding eigenfunction ηǫ is positive and belongs to C2([ǫ,1/ǫ]). We choose it so that
Z 1/ǫ
ǫ
ηǫ(x)2dµ(x) = 1. (1)
Let us recall some results of [22]:
Theorem (Pinsky (1985)) The Yaglom limit associated with Pǫ exists for all initial distributions δx, x ∈]ǫ,1/ǫ[, and doesn’t depend on x. This limit is a QSD, which we denote by νǫ. Furthermore, we have
dνǫ(x) = ηǫ(x)dµ(x) R1/ǫ
ǫ ηǫ(x)dµ(x). (2)
In fact, νǫ is the unique QSD of the process, as proved in Lemma 19 below, but we won’t use it in this section. The aim of this section is to study the asymptotic behaviour of (νǫ) when ǫ goes to 0.
From now,M1(]0,+∞[) denotes the space of probability measures on ]0,+∞[ equipped with the weak topology. The following hypotheses arise naturally in the proof of Theorem 1, which is based on a compactness-uniqueness method.
Hypothesis 1 (H1) W is bounded below by −C, where C is a positive constant. More- over, W(x)→+∞ when x→ ∞.
Hypothesis 2 (H2) Z +∞
1
e−2Q(x)dx <+∞ and Z 1
0
1
W(x) +C+ 1µ(dx)<+∞. Hypothesis 3 (H3)
Z +∞
1
e−2Q(x)dx <+∞ and Z 1
0
x e−Q(x)dx <+∞.
Theorem 1 Assume that hypotheses (H1) and (H2 or H3) are satisfied. Then νǫ−→ǫ→0ν ∈ M1(]0,+∞[),
where ν is a QSD for P0, which is equal to the Yaglom limit limt→+∞P0
x(Xt∈.|t < τ∂),
∀x∈]0,+∞[.
Remark 1 The hypotheses (H1) and (H2 or H3) are the assumptions that are made in [3] to prove the existence of the Yaglom limit.
Remark 2 If a process satisfies the hypotheses of Theorem 1, then it is killed in finite time a.s or it is never killed a.s. Indeed, assume that the process can be killed in finite time with a positive probability. Then R0
1 eQ(x) Rx
1 e−Q(y)dy
dx < +∞ (see [14]) and R+∞
1 eQ(x)dx= +∞ (as a consequence of (H1) and (H2 or H3)). But this two conditions are fulfilled if and only if the process is killed in finite time almost surely (see [14, Theorem 3.2 p.450]).
Remark 3 The existence of a QSD for P0 can be seen as a consequence of Theorem 1.
The existence of the Yaglom limit is proved in [3, Theorem 5.2].
Remark 4 In Part 2.3, we give the counterpart of Theorem 1 for diffusions with values in a bounded interval.
The end of the section is devoted to the proof of Theorem 1.
2.1 Tightness of the family (νǫ)0<ǫ<1/2
This part is devoted to the proof of the following result,
Proposition 2 Assume that the hypotheses (H1) and (H2 or H3) are satisfied. Then the family (νǫ)0<ǫ≤1/2 is tight. Moreover, every limit point is absolutely continuous with respect to the Lebesgue measure.
We know thatLǫηǫ=−λǫηǫ,ηǫ ∈C2([ǫ,1/ǫ]), andηǫ satisfies the differential equation 1/2ηǫ′′(x)−q(x)η′ǫ(x) =−λǫηǫ(x)
with the boundary conditions
ηǫ(ǫ) =ηǫ(1/ǫ) = 0.
Define vǫ =ηǫe−Q. By (1), we know that:
Z 1/ǫ
ǫ
vǫ(x)2dx= 1.
We have
vǫ(x)W(x)−vǫ′′(x) = 2λǫvǫ(x), with the boundary conditions
vǫ(ǫ) =vǫ(1/ǫ) = 0. (3)
Lemma 3 Assume that the hypothesis (H1) is fulfilled. Then (vǫ)0<ǫ<1/2 is uniformly bounded above and the family (vǫ2(x)dx)0<ǫ<1/2 is tight.
Proof of Lemma 3 : From the differential equation satisfied by vǫ, we have vǫ(x)2W(x)−vǫ′′(x)vǫ(x) = 2λǫvǫ(x)2.
Integrating by parts and looking at the boundary conditions (3), Z 1/ǫ
ǫ
vǫ′′(x)vǫ(x)dx=− Z 1/ǫ
ǫ
vǫ′(x)2dx, where vǫ is normalized in L2(dx). That implies
Z 1/ǫ
ǫ
vǫ′(x)2dx+ Z 1/ǫ
ǫ
vǫ(x)2W(x)dx = 2λǫ.
The eigenvalue λǫ of −Lǫ is given by (see for instance [28, chapter XI, part 8]) λǫ = inf
φ∈C∞0 (]ǫ,1/ǫ[)(Lǫφ,φ)µ,
= inf
φ∈C∞0 (]ǫ,1/ǫ[)(L0φ,φ)µ, (4)
where C0∞(]ǫ,1/ǫ[) is the vector space of infinitely differentiable functions with compact support in ]ǫ,1/ǫ[ and (f,g)µ =R+∞
0 f(u)g(u)dµ(u). We deduce from it that λǫ increases with ǫ and is uniformly bounded above by λ1/2.
We have then 0≤
Z 1/ǫ
ǫ
vǫ′(x)2dx+ Z 1/ǫ
ǫ
vǫ(x)2(W(x) +C+ 1)dx≤2λ1/2+C+ 1. (5) Looking at the boundary conditions (3), we obtain, for all x∈]ǫ,1/ǫ[,
vǫ2(x) = −2 Z 1/ǫ
x
vǫ′(y)vǫ(y)dy
≤ − 2
min[x,1/ǫ[
√W +C+ 1 Z 1/ǫ
x
vǫ′(y)vǫ(y)p
W(y) +C+ 1dy.
Then, applying the Cauchy-Schwarz inequality to the right term above,
vǫ2(x) ≤ 2
min[x,1/ǫ[√
W +C+ 1
sZ 1/ǫ x
vǫ′(y)2dy
sZ 1/ǫ x
vǫ(y)2(W(y) +C+ 1)dy.
From (5), the integral product is bounded by 2λ1/2 +C+ 1, thus ∃A > 0, independent from ǫ, such that
vǫ2(x) ≤ A
min[x,1/ǫ[√
W +C+ 1
≤ A
min[x,+∞[√
W+C+ 1, (6)
where W(x) +C+ 1 ≥ 1 for all x ∈]0,+∞[, thanks to Hypothesis (H1). That implies the first part of Lemma 3.
Let us prove that the family (vǫ2dx)0<ǫ<1/2 is tight. Fix δ > 0. We have to find a compact subset Kδ in ]0,+∞[ such that
Z
]0,+∞[\Kδ
vǫ2(x)dx ≤δ, (7)
for all ǫ∈]0,1/2[. Thanks to (6), we have vǫ2 ≤A, then Z δ/(2A)
0
vǫ2 ≤δ/2.
From the second part of Hypothesis (H1), ∃Mδ > 0 such that W(x) +C + 1 > 2(2λ+ C+ 1)/δ for all x≥Mδ. That implies
Z +∞
Mδ
vǫ(x)2dx ≤ Z 1/ǫ
ǫ
vǫ(x)2δ(W(x) +C+ 1) 2(2λ+C+ 1) dx
≤ δ/2,
where the last inequality is due to (5). Finally, the compact set Kδ= [δ/(2A),Mδ] satisfies (7). 2
Lemma 4 Assume that (H1) is satisfied. Then R1/ǫ
ǫ ηǫ(y)dµ(y) is uniformly bounded below by a constant B >0.
Proof of Lemma 4 : Assume thatR1/ǫ
ǫ ηǫ(y)dµ(y) isn’t uniformly bounded below : one can find a sub-sequence R1/ǫk
ǫk vǫk(y)e−Q(y)dy = R1/ǫk
ǫk ηǫk(y)dµ(y), where ǫk → 0, which tends to 0. From Lemma 3, (vǫ)0<ǫ≤1/2 is uniformly bounded, so thatR1/ǫk
ǫk vǫk(y)2e−Q(y)dy→0.
The family (vǫ(x)2dx) being tight, one can find (after extracting a sub-sequence) a positive map m such that, for all continuous and boundedφ :R+→R,
Z 1/ǫk
ǫk
vǫk(y)2φ(y)dy→ Z +∞
0
m(y)φ(y)dy. (8)
Indeed, (v2ǫ) being uniformly bounded, all limit measure is absolutely continuous with respect to the Lebesgue measure. In particular,
Z 1/ǫk
ǫk
vǫk(y)2min (e−Q(y),1)dy → Z +∞
0
m(y) min (e−Q(y),1)dy,
then Z +∞
0
m(y) min (e−Q(y),1)dy= 0.
But min (e−Q(.),1) is continuous and positive on R+, so that m vanishes almost every where. Finally, by the convergence property (8) applied toφequal to 1 almost everywhere, we have
1 = Z 1/ǫk
ǫk
vǫ2kdx→0, what is absurd. Thus, one can define B = infǫR1/ǫ
ǫ ηǫ(y)dµ(y)/A >0. 2
Lemma 5 Assume that (H1) and (H2) are satisfied. Then the family(ηǫ(x)dµ(x))0<ǫ<1/2
is tight.
Proof of Lemma 5 : By (5), we have Z 1/ǫ
ǫ
ηǫ2(y)(W(y) +C+ 1)dµ(y) =
Z 1/ǫ
ǫ
vǫ2(y)(W(y) +C+ 1)dy
≤ λ1/2 +C+ 1.
For all δ,M >0, using Cauchy-Schwarz inequality, we get on one hand Z δ
0
ηǫ(y)dµ(y) ≤
Z δ 0
ηǫ(y)2(W(y) +C+ 1)dµ(y)
12 Z δ 0
1
W(y) +C+ 1dµ(y) 12
(9)
≤ λ1/2+C+ 112 Z δ 0
1
W(y) +C+ 1dµ(y) 12
. (10)
On the other hand, Z +∞
M
ηǫ(y)dµ(y) ≤
Z +∞
M
ηǫ2(y)dµ(y)
12 Z +∞
M
dµ(y) 12
(11)
≤
Z +∞
M
dµ(y) 12
. (12)
Thanks to (H2), both terms are going to 0 uniformly inǫ, whenδ andM tend respectively to 0 and +∞. As a consequence, the family (ηǫ(x)dµ(x))0<ǫ<1/2 is tight. 2
Lemma 6 Assume that (H1) and (H3) hold. Then the family(ηǫ(x)dµ(x))0<ǫ<1/2 is tight.
Proof of Lemma 6 : The first part of the hypothesis (H3) is the same as (H2)’s one, then Z +∞
M
ηǫ(y)dµ(y)→0 when M goes to infinity, uniformly in ǫ.
Moreover, there exists a constantK >0 such that, for anyx∈]0,1] and anyǫ∈]0,1/2], ηǫ(x)≤KxeQ(x).
This is a consequence of [3, Proposition 4.3] whose proof is still available under our settings. This inequality allows us to conclude the proof of Lemma 6. 2
Thanks to equality (2) and Lemmas 4, 5 and 6, the first part of Proposition 2 is proved.
Moreover, νǫ has a density with respect to the Lebesgue measure which is bounded on every compact set, uniformly in ǫ > 0. Thus every limit point is absolutely continuous with respect to the Lebesgue measure.
2.2 The limit points of the family (νǫ)0<ǫ<1/2
Proposition 7 Assume that Hypotheses (H1) and (H2 or H3) are fulfilled and let ν be a probability measure which is the limit of a sub-sequence (νǫk)k∈N, where ǫk → 0 when k → ∞. Then ν is a QSD with respect to P0.
Proof of Proposition 7 : From Proposition 2, the family (νǫ)0<ǫ<1/2 is tight. Let ν be a limit point of the family (νǫ)0<ǫ<1/2. There exists a sub-sequence (νǫk)k which converges to ν, where (ǫk)k∈N is a decreasing sequence which tends to 0. We already know that ν is absolutely continuous with respect to the Lebesgue measure. That implies that, for all open intervals D=]c,d[⊂R+,
νǫk(D)→ν(D), (13)
and, for all bounded maps φ continuous on R+, Z
R+
φ(x)dνǫk(x)→ Z
R+
φ(x)dν(x). (14)
Letνt (resp. νǫ,t) be the distribution at time t of a diffusion with law P0ν (resp. Pǫν
ǫ), conditioned to be not killed until time t, that is
νt(dx) =P0ν(ωt∈dx|τ∂ > t) and
νǫ,t(dx) =Pǫν
ǫ(ωt∈dx|τ∂ > t)
The probability measure νǫ being a QSD for Pǫ, we have νǫ,t =νǫ for all t >0. We want to show that ν =νt for all t >0, we have then to prove the following convergence result:
∀t >0, ∀D=]c,d[⊂R+, νǫk,t(D) −→
k→+∞νt(D). (15)
Indeed, suppose that (15) holds, then on the one hand, νǫk(D) = νǫk,t(D) → νt(D). In the other hand νǫk(D) → ν(D). We have then νt(]c,d[) = ν(]c,d[), ∀]c,d[⊂ R+ and this conclude the proof of Proposition 7.
Let us prove (15). By definition, νǫk,t(D) =
R
]ǫk,1/ǫk[Pǫxk(ωt∈D)dνǫk(x) R
]ǫk,1/ǫk[Pǫxk(τ∂ > t)dνǫk(x) . The numerator is equal to
Z
]ǫk,1/ǫk[
Pǫk
x (ωt∈D)dνǫk(x) = Z
]ǫk,1/ǫk[
P0
x(ωt ∈D)dνǫk(x) +
Z
]ǫk,1/ǫk[
Pǫk
x(ωt ∈D)−P0
x(ωt∈D)
dνǫk(x) For all t > 0, the map x 7→ P0x(ωt ∈ D) is continuous and bounded, then, by the convergence property (14),
Z
]ǫk,1/ǫk[
P0x(ωt ∈D)dνǫk(x)→ Z
]0,∞[
P0x(ωt∈D)dν(x),
Assume that (H2) is fulfilled, then, similarly to (9) and (11), we have, for all bounded continuous functions f :]0,+∞[→R and allM > 0,
Z +∞
M
f(x)ηǫk(x)dµ(x)≤
Z +∞
M |f(x)|2dµ(x) 12
, and, for all m >0,
Z m
0
f(x)ηǫk(x)dµ(x)≤ λ1/2 +C+ 112 Z m 0
|f(x)|2
W(x) +C+ 1dµ(x)
!12 . Replacing f(x) byP0
x(ωt∈D)−Pǫk
x (ωt ∈D), which is decreasing to 0 whenk → ∞, and by monotone convergence theorem, we have
Z
]0,m[
Pǫk
x (ωt∈D)−P0
x(ωt ∈D)
dνǫk(x) −→
k→+∞0
and Z
]M,+∞[
Pǫk
x(ωt∈D)−P0
x(ωt∈D)
dνǫk(x) −→
k→+∞0.
Finally, the density of νǫk being bounded above in every compact set [m,M], uniformly in ǫk, the same argument of monotone convergence gives us
Z
]ǫk,1/ǫk[
Pǫk
x (ωt∈D)−P0
x(ωt∈D)
dνǫk(x) −→
k→+∞0.
With similar arguments, the same holds under (H3). Finally, we obtain Z
]ǫk,1/ǫk[
Pǫk
x (ωt∈D)dνǫk(x)→ Z
]0,+∞[
P0
x(ωt∈D)dν(x).
Thanks to [3, Lemma 5.3 and Theorem 2.3], the map x 7→ P0
x(τ∂ > t) = P0
x(ωt ∈ ]0,+∞[) is continuous, and P0
x(τ∂ > t)−Pǫk
x(τ∂ > t) is increasing to 0 when k → ∞. Thus the denominator can be treated in the same way. 2
We can now conclude the proof of Theorem 1:
Proposition 8 Assume that (H1) and (H2 or H3) hold. The limit measure ν in the statement of Proposition 7 is unique. Moreover ν is the Yaglom limit associated withP0
x,
∀x∈]0,+∞[.
Proof of Proposition 8 : The proof of Proposition 7 implies that Pǫ
νǫ(τ∂ > t)−→ǫ→0P0
ν(τ∂ > t), ∀t >0.
The probability measure νǫ being a QSD forPǫ, we have Pǫ
νǫ(τ∂ > t) =e−λǫt,∀t >0.
Thanks to (4), λǫ is decreasing to λ0 = infφ∈C0∞(]0,+∞[)(L0φ,φ)µ when ǫ goes to 0. As a consequence,
P0
ν(τ∂ > t) =e−λ0t,∀t >0.
In this case, the density of ν with respect to dµis an eigenfunction of L0 with eigenvalue
−λ0 <0, where (L0)∗ is the adjoint operator of L0 (this is a consequence of the spectral decomposition proved in [3, Theorem 3.2]). As defined, −λ0 is at the bottom of the spectrum of (L0)∗. Thanks to [3, Theorem 3.2], this eigenvalue is simple. Moreover, [3, Theorem 5.2] states that this QSD is equal to limt→+∞P0
x(Xt∈.|τ∂ > t), what concludes the proof. 2
2.3 Diffusions with values in a bounded interval
Theorem 1 is stated for 1-dimensional diffusions with values in ]0,+∞[. However, most of the proofs can be easily adapted to diffusions with values in a bounded interval ]a,b[, where −∞< a < b <+∞, defined by the SDE
dXt=dBt−q(Xt)dt, X0 =x∈]a,b[,
and killed when it hits a or b. Here B is a standard Brownian motion and q ∈C1(]a,b[).
More precisely, let us denote byP0 the law of such a diffusion. For each ǫ >0, define Pǫ as the law of a diffusion with values in ]a+ǫ,b−ǫ[, driven by the SDE
dXt=dBt−q(Xt)dt, X0 =x∈]a+ǫ,b−ǫ[,
and killed when it hits a+ǫ or b −ǫ. As proved in [22], there exists a unique QSD νǫ
associated with Pǫ.
We define Q(x) = Rx
(a+b)/2q(y)dy and W(x) = q(x)2 − q′(x). The counterpart of Theorem 1 under these settings is
Theorem 9 Assume that the following hypotheses are fulfilled:
Hypothesis 4 (HH1) W is uniformly bounded below by −C, where C is a positive constant.
Hypothesis 5 (HH2) x 7→ W(x)+C+11 e−2Q(x) or x 7→ (x−a)e−Q(x) is integrable on a neighbourhood of a.
Hypothesis 6 (HH3) x 7→ W(x)+C+11 e−2Q(x) or x 7→ (b −x)e−Q(x) is integrable on a neighbourhood of b.
Then the family of QSD(νǫ)is tight as family of measures on ]a,b[. Moreover, every limit point of the family (νǫ)0<ǫ<1/2 is a QSD for P0.
Remark 5 Our aim isn’t to develop this part, but we point out that to show that Propo- sition 8 remains valid, most of the arguments used to prove the key results [3, Theorem 3.2] and [3, Theorem 5.2] can be adapted to these settings.
3 Approximation of νǫ, QSD for Pǫ
We are interested in proving an approximation method for the QSD associated with Pǫ. It will be sufficient to prove it for any diffusion (Xt) taking its values in ]0,1[ in place of ]ǫ,1/ǫ[, defined by the stochastic differential equation (SDE)
dXt=dBt−q(Xt)dt, X0 =x >0, (16) and killed when Xt hits the boundary {0,1}. Here B is a real Brownian motion and q ∈C1([0,1]). The law of X will be denoted byP.
From [22], the QSD ofX is unique and equals the Yaglom limit. It will be denoted by ν. For notational convenience, new notations have been defined for this section, which is totally independent of the previous one.
FixN ≥2 and let us define formally the interacting particle process with N particles described in the introduction. Let B1,...,BN be N independent Brownian motions and (X01,...,X0N)∈]0,1[N be the starting point of the process.
For each i ∈ {1,...,N}, the particle Xi evolves in ]0,1[ and satisfies the SDE dXti = dBti−q(Xti)dt(and then it is independent of the others) until τ1i = inf{t ≥0, Xt−i = 0 or 1}.
At time τ1 = min{τ11,...,τN1}, the path of a particle, denoted byi1 (it is unique), has a left limit equal to 0 or 1.
A particle j1 is chosen in {1,...,N} \ {i1}. The particlei1 jumps on the position of the particle j1: we set Xτi11 :=Xτj11.
After time τ1, each particle Xi evolves in ]0,1[ with respect to the SDE dXti = dBti−q(Xti)dtuntil τ2i = inf{t > τ1, Xt−i = 0 or 1}. At time τ1, all the particles are in ]0,1[, so that we have τ2i > τ1 for all i∈ {1,...,N} almost surely.
At timeτ2 = min{τ21,...,τ2N}(which is then strictly bigger thanτ1), a unique particle i2 has a path whose left limit is equal to 0 or 1.
A particle j2 is chosen in {1,...,N} \ {i2}. The particlei2 jumps on the position of the particle j2: we set Xτi22 :=Xτj22.
After time τ2, the particles evolve independently from each other and so on.
Following this way, we define the strictly increasing sequence of stopping times 0 < τ1 <
τ2 < τ3 < ..., the time τ∞ = limn→∞τn and the interacting particle system (Xt1,...,XtN) for all t∈[0,τ∞[. The law of (X1,...,XN) will be denoted by Pipp.
